Energy release rate and stress intensity factors in planar elasticity in presence of smooth cracks

  • Stefano AlmiEmail author
  • Ilaria Lucardesi


In this work we first analyze the singular behavior of the displacement u of a linearly elastic body in dimension 2 close to the tip of a smooth crack, extending the well-known results for straight fractures to general smooth ones. As conjectured by Griffith (Phys Eng Sci 221:163–198, 1921), u behaves as the sum of an \(H^{2}\)-function and a linear combination of two singular functions whose profile is similar to the square root of the distance from the tip. The coefficients of the linear combination are the so called stress intensity factors. Afterwards, we prove the differentiability of the elastic energy with respect to an infinitesimal fracture elongation and we compute the energy release rate, enlightening its dependence on the stress intensity factors.


Brittle fracture Stress intensity factor Energy release rate 

Mathematics Subject Classification

35Q74 35B40 74R10 74G40 74G70 



The work of the authors was partially supported by the European Research Council under Grant No. 501086 “High-Dimensional Sparse Optimal Control”. S.A. also acknowledges the support of the SFB TRR109.


  1. 1.
    Almi, S.: Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening. ESAIM Control Optim. Calc. Var. 23, 791–826 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Almi, S.: Quasi-static hydraulic crack growth driven by Darcy’s law. Adv. Calc. Var. 11, 161–191 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Babadjian, J.-F., Chambolle, A., Lemenant, A.: Energy release rate for non-smooth cracks in planar elasticity. J. Éc. Polytech. Math. 2, 117–152 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all’interno, Quaderni, Scuola Normale Superiore Pisa, Pisa (1980)Google Scholar
  5. 5.
    Chambolle, A., Francfort, G.A., Marigo, J.-J.: Revisiting energy release rates in brittle fracture. J. Nonlinear Sci. 20, 395–424 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chambolle, A., Lemenant, A.: The stress intensity factor for non-smooth fractures in antiplane elasticity. Calc. Var. Partial Differ. Equ. 47, 589–610 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dal Maso, G., Orlando, G., Toader, R.: Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length. NoDEA Nonlinear Differ. Equ. Appl. 22, 449–476 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal. 186, 477–537 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Francfort, G.A., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Griffith, A.A.: The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Math. Phys. Eng. Sci. 221, 163–198 (1921)CrossRefGoogle Scholar
  11. 11.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Elliptic Problems in Nonsmooth Domains, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)zbMATHGoogle Scholar
  12. 12.
    Grisvard, P.: Singularities in Boundary Value Problems. Research in Applied Mathematics, vol. 22. Springer, Berlin (1992)zbMATHGoogle Scholar
  13. 13.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  14. 14.
    Knees, D., Mielke, A.: Energy release rate for cracks in finite-strain elasticity. Math. Methods Appl. Sci. 31, 501–528 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kondrat’ev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč. 16, 209–292 (1967)MathSciNetGoogle Scholar
  16. 16.
    Kozlov, V.A., Mazya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Elliptic Boundary Value Problems in Domains with Point Singularities, vol. 52. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  17. 17.
    Kozlov, V.A., Mazya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)Google Scholar
  18. 18.
    Lazzaroni, G., Toader, R.: Energy release rate and stress intensity factor in antiplane elasticity. J. Math. Pures Appl. 95, 565–584 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lazzaroni, G., Toader, R.: A model for crack propagation based on viscous approximation. Math. Models Methods Appl. Sci. 21, 2019–2047 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Negri, M., Toader, R.: Scaling in fracture mechanics by Bažant law: from finite to linearized elasticity. Math. Models Methods Appl. Sci. 25, 1389–1420 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zehnder, A.T.: Fracture Mechanics. Lecture Notes in Applied and Computational Mechanics, vol. 62. Springer, London (2012)Google Scholar

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Authors and Affiliations

  1. 1.TU MünchenGarching bei MünchenGermany
  2. 2.Institut Élie Cartan de LorraineVandoeuvre-lès-NancyFrance

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